Imagine you are new to baby-sitting. You are tasked to observe two naughty toddlers in a room- to see how they move around and understand what they talk to each other. Somehow, you are managing to map out the goldfish kids. Next into the scene comes another kid. Although your tasks remain the same, the third kid has put you in a tougher role. You are now baffled.

If you swap the example of three kids with three celestial bodies, you are none but a struggling scientist looking at the famous three-body problem. Say, the Earth, the Moon, and a satellite.

In physics, the three-body problem is understanding how three masses will behave over time, given their behaviour at an instant. In the case of Earth, the satellite, and the moon, the job is to understand what path they will follow in the future. To look at it, we are provided with their initial position and velocity. Here, a conventional, precise solution fails to exist due to the complexities involved in the interaction of the three masses.

A way to bypass the complexity is to ignore the satellite’s effect on the Earth and the moon. This sounds sensible as the latter masses are huge compared to a tiny satellite which is like the third kid who arrived in the room but unlike the first two, is fortunately quiet. Then you may even ignore the third kid and just focus on the first two.

Even if the satellite is puny enough to cause any significant effect on Earth and the moon, according to physicists, the problem still remains complex to have an exact solution to the equation of motion. Any tiny changes in initial conditions lead to wildly different outcomes. Hence, long-term predictions become impossible.

The mathematician Henri Poincaré showed that even when reduced to a system with two degrees of freedom (read: neglect the third quiet kid in the hall), there’s no complete set of energy or momentum to predict its behaviour fully.

But Poincaré discovered the secret sauce to solve this!

The mathematician found out that, instead of relying on conventional mathematical tools like integration and differentiation, geometric representations of the possible energy states could be helpful. This representation is formally known as a phase space diagram. As Poincaré discovered, the phase space beautifully packs stable regions among chaotic zones, hence partly revealing a solution to the problems posed by the complexities.

Poincaré's work pointed out that the phase space (mathematical representation of all possible energy states) has a hidden geometric structure. Now, the study of these geometric structures is known as Symplectic geometry.

Nonetheless, the solution to the problem required the efforts of many mathematicians who studied enormous surfaces and spent years solving it. “Solving any mathematical problem is not a question of one strange mathematician thinking and solving it. It requires effort from many people,” said Yakov Eliashberg, Professor of Mathematics and winner of the 2020 Wolf Prize in Mathematics, at the TNQ Distinguished Lecture Series 2024. He states that symplectic methods are pertinent to understanding real-world systems, such as planetary bodies, where exact solutions are analytically elusive.

To solve problems like the three-body problem, understanding the interplay between geometry and dynamic systems is important. A tiny error could drastically set the satellite off the desired track, even if there's only a slight miscalculation. Poincaré observed that such intricate systems have orderly and chaotic motions intertwined. The symplectic method allows scientists to simulate the system more reliably, ensuring energy conservation and stability even in chaos. It's a geometric tool sought out when traditional calculus fails to fetch exact solutions to problems in classical mechanics, for instance, the three-body problem.

No matter how the system evolves, this structure is preserved. For instance, in a system with two degrees of freedom, if there's a periodic orbit, the symplectic structure ensures that the area on a cross-section in a phase space remains unchanged even when moving from one periodic orbit to the other as the system changes energy due to many factors.

In the real world, perfectly periodic solutions are rare. Nonetheless, tracking the collection of partially periodic orbits provides a stable reference around which we can analyse the behaviour of nearby trajectories in the phase space. Poincaré’s work, called a return map, tracks how points near a periodic orbit evolve after each cycle. The return map shows that the geometric structure is unchanged even if the system is chaotic over time.

Mathematician Vladimir Arnold further cemented Poincaré’s work. In 1965, Arnold revealed that an area preserving a geometric structure must have at least as many fixed points as a smooth function has critical points on that surface. In the 1970s, Eliashberg provided the Arnold conjecture for 2D surfaces. He proved that symplectic maps must have fixed points, such as critical points, on a smooth landscape despite the change in geometric structure. However, mathematicians like Mikhael Gromov made it seem that symplectic geometry fails to be preserved in higher dimensions. In 1980, Eliashberg and others proved that symplectic systems remain intact even in higher dimensions. This work gave rise to the Eliashberg-Gromov theorem.

The theorem is the cornerstone of symplectic topology, revealing hidden order in chaotic systems. For example, in fluid flow.

Eliashberg also laid the groundwork for symplectic field theory (SFT) which studies the behaviour of curves and surfaces in spaces where mechanics and geometry intertwine. The theory focuses on how periodic orbits split over time as the system gets into chaos. It’s a tool to tackle problems from classical mechanics to string theory.

Symplectic field theory reveals how complex systems such as planetary motions encode hidden patterns in their chaotic behaviour- order amidst chaos!